Analysis and Geometry on Complex Homogeneous Domains

by Faraut, Jacques; Kaneyuki, Soji; Koranyi, Adam; Lu, Qi-Keng; Roos, Guy

• ISBN13: 9780817641382
• ISBN10: 0817641386
• Format: Hardcover
• Copyright: 1999-08-01
• Publisher: Birkhauser

Summary

This book, authored by five experts in their respective fields, covers a number of important areas in complex analysis and geometry. Unfolding the subjects from the basics to the more complex, the book gives a rapid paced and efficient exposition, without compromising proofs and examples. Useful as a self-study resource, it will guide readers through the subject considerably faster and more efficiently than competing books.

Table of Contents

Preface	xi
Part I Function Spaces on Complex Semi-groups	1	(102)
Jacques Faraut
Introduction	3	(2)
Hilbert Spaces of Holomorphic Functions	5	(14)
Reproducing kernels	5	(10)
Invariant Hilbert spaces of holomorphic functions	15	(4)
Invariant Cones and Complex Semi-groups	19	(14)
Complex semi-groups	19	(2)
Invariant cones in a representation space	21	(5)
Invariant cones in a simple Lie algebra	26	(7)
Positive Unitary Representations	33	(12)
Self-adjoint operators	33	(5)
Unitary representations	38	(3)
Positive unitary representations	41	(4)
Hilbert Function Spaces on Complex Semi-groups	45	(20)
Schur orthogonality relations	45	(8)
The Hardy space of a complex semi-group	53	(6)
The Cauchy-Szego kernel and the Poisson kernel	59	(3)
Spectral decomposition of the Hardy space	62	(3)
Hilbert Function Spaces on SL(2, C)	65	(18)
Complex Olshanski semi-group in SL(2, C)	65	(2)
Irreducible positive unitary representations	67	(6)
Characters and formal dimensions of the representations πm	73	(3)
Bi-invariant Hilbert spaces of holomorphic functions	76	(2)
The Hardy space	78	(1)
The Bergman space	79	(4)
Hilbert Function Spaces on a Complex Semi-simple Lie Group	83	(20)
Bounded symmetric domains	83	(5)
Irreducible positive unitary representations	88	(8)
Characters and formal dimensions	96	(2)
Bi-invariant Hilbert spaces of holomorphic functions	98	(5)
References	99	(4)
Part II Graded Lie Algebras and Pseudo-hermitian Symmetric Spaces	103	(80)
Soji Kaneyuki
Introduction	105	(2)
Semisimple Graded Lie Algebras	107	(20)
Root theory of real semisimple Lie algebras	107	(4)
Semisimple graded Lie algebras	111	(5)
Example	116	(3)
Tables	119	(8)
Symmetric R-Spaces	127	(24)
Symmetric R-spaces and their noncompact duals	127	(6)
Sylvester's law of inertia in simple GLA's	133	(8)
Generalized conformal structures and causal structures	141	(10)
Pseudo-Hermitian Symmetric Spaces	151	(32)
Pseudo-Hermitian spaces and nonconvex Siegel domains	151	(15)
Simple reducible pseudo-Hermitian symmetric spaces	166	(17)
References	179	(4)
Part III Function Spaces on Bounded Symmetric Domains	183	(100)
Adam Koranyi
Introduction	185	(2)
Bergman Kernel and Bergman Metric	187	(6)
Domains in Cn	187	(1)
Bergman kernel, reproducing kernels	188	(2)
The Bergman metric	190	(3)
Symmetric Domains and Symmetric Spaces	193	(10)
Basic facts, definitions	193	(1)
Riemannian symmetric spaces	194	(2)
Theory of oiLa's	196	(1)
OiLa's of bounded symmetric domains	197	(3)
Cartan subalgebras	200	(3)
Construction of the Hermitian Symmetric Spaces	203	(8)
The Borel imbedding theorem	203	(2)
The Harish-Chandra realization	205	(4)
Remarks on classification	209	(2)
Structure of Symmetric Domains	211	(14)
Restricted root system, boundary orbits	211	(6)
Decomposition under the Cayley transform	217	(8)
The Weighted Bergman Spaces	225	(18)
Analysis on symmetric domains	225	(5)
Decomposition under K	230	(4)
Spaces of holomorphic functions	234	(9)
Differential Operators	243	(14)
Properties of δs	243	(3)
Invariant differential operators on Ω	246	(2)
Further results on D(Ω)	248	(3)
Extending Dα to p+	251	(6)
Function Spaces	257	(26)
The holomorphic discrete series	257	(2)
Analytic continuation of the holomorphic discrete series	259	(5)
Explicit formulas for the inner products	264	(3)
Lq-spaces and Bergman type projections	267	(3)
Some questions of duality	270	(4)
Further results	274	(9)
References	277	(6)
Part IV The Heat Kernels of Non Compact Symmetric Spaces	283	(142)
Qi-keng Lu
Introduction	285	(18)
The Laplace-Beltrami Operator in Various Coordinates	303	(18)
The Integral Transformations	321	(16)
The Heat Kernel of the Hyperball RR (m, n)	337	(14)
The Harmonic Forms on the Complex Grassmann Manifold	351	(14)
The Horo-hypercircle Coordinate of a Complex Hyperball	365	(16)
The Heat Kernel of RII(m)	381	(12)
The Matrix Representation of NIRGSS	393	(32)
References	423	(2)
Part V Jordan Triple Systems	425	(110)
Guy Roos
Introduction	427	(2)
Polynomial Identities	429	(22)
Definition of Jordan triple systems	429	(2)
Identities of minimal degree	431	(4)
Jordan representations and duality	435	(5)
The fundamental identity of degree 7	440	(1)
The Bergman operator	441	(10)
Jordan Algebras	451	(10)
Jordan algebras arising from a JTS	451	(1)
Identities in a Jordan algebra	452	(6)
The JTS associated to a Jordan algebra	458	(3)
The Quasi-inverse	461	(14)
Quasi-invertibility in a Jordan algebra	461	(5)
Quasi-invertibility in a JTS	466	(3)
Identities for the quasi-inverse	469	(1)
Differential equations	470	(2)
Addition formulas	472	(3)
The Generic Minimal Polynomial	475	(22)
Unital Jordan algebras	475	(11)
General Jordan algebras	486	(2)
Jordan triple systems	488	(9)
Tripotents and Peirce Decomposition	497	(10)
Tripotent elements	497	(1)
Peirce decomposition	498	(3)
Orthogonality of tripotents	501	(2)
Simultaneous Peirce decomposition	503	(4)
Hermitian Positive JTS	507	(22)
Positivity	507	(3)
Spectral decomposition	510	(8)
Automorphisms	518	(5)
The spectral norm	523	(2)
Classification of Hermitian positive JTS	525	(4)
Further Results and Open Problems	529	(6)
Schmid decomposition	529	(1)
Compactification of an hermitian positive JTS	530	(1)
Projective imbedding	531	(1)
Volume computations	531	(3)
Some open problems	534	(1)
References	535	(2)
Index	537